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Mirrors > Home > ILE Home > Th. List > f00 | GIF version |
Description: A class is a function with empty codomain iff it and its domain are empty. (Contributed by NM, 10-Dec-2003.) |
Ref | Expression |
---|---|
f00 | ⊢ (𝐹:𝐴⟶∅ ↔ (𝐹 = ∅ ∧ 𝐴 = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffun 5048 | . . . . 5 ⊢ (𝐹:𝐴⟶∅ → Fun 𝐹) | |
2 | frn 5052 | . . . . . . 7 ⊢ (𝐹:𝐴⟶∅ → ran 𝐹 ⊆ ∅) | |
3 | ss0 3257 | . . . . . . 7 ⊢ (ran 𝐹 ⊆ ∅ → ran 𝐹 = ∅) | |
4 | 2, 3 | syl 14 | . . . . . 6 ⊢ (𝐹:𝐴⟶∅ → ran 𝐹 = ∅) |
5 | dm0rn0 4552 | . . . . . 6 ⊢ (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅) | |
6 | 4, 5 | sylibr 137 | . . . . 5 ⊢ (𝐹:𝐴⟶∅ → dom 𝐹 = ∅) |
7 | df-fn 4905 | . . . . 5 ⊢ (𝐹 Fn ∅ ↔ (Fun 𝐹 ∧ dom 𝐹 = ∅)) | |
8 | 1, 6, 7 | sylanbrc 394 | . . . 4 ⊢ (𝐹:𝐴⟶∅ → 𝐹 Fn ∅) |
9 | fn0 5018 | . . . 4 ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅) | |
10 | 8, 9 | sylib 127 | . . 3 ⊢ (𝐹:𝐴⟶∅ → 𝐹 = ∅) |
11 | fdm 5050 | . . . 4 ⊢ (𝐹:𝐴⟶∅ → dom 𝐹 = 𝐴) | |
12 | 11, 6 | eqtr3d 2074 | . . 3 ⊢ (𝐹:𝐴⟶∅ → 𝐴 = ∅) |
13 | 10, 12 | jca 290 | . 2 ⊢ (𝐹:𝐴⟶∅ → (𝐹 = ∅ ∧ 𝐴 = ∅)) |
14 | f0 5080 | . . 3 ⊢ ∅:∅⟶∅ | |
15 | feq1 5030 | . . . 4 ⊢ (𝐹 = ∅ → (𝐹:𝐴⟶∅ ↔ ∅:𝐴⟶∅)) | |
16 | feq2 5031 | . . . 4 ⊢ (𝐴 = ∅ → (∅:𝐴⟶∅ ↔ ∅:∅⟶∅)) | |
17 | 15, 16 | sylan9bb 435 | . . 3 ⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → (𝐹:𝐴⟶∅ ↔ ∅:∅⟶∅)) |
18 | 14, 17 | mpbiri 157 | . 2 ⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → 𝐹:𝐴⟶∅) |
19 | 13, 18 | impbii 117 | 1 ⊢ (𝐹:𝐴⟶∅ ↔ (𝐹 = ∅ ∧ 𝐴 = ∅)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 = wceq 1243 ⊆ wss 2917 ∅c0 3224 dom cdm 4345 ran crn 4346 Fun wfun 4896 Fn wfn 4897 ⟶wf 4898 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-fun 4904 df-fn 4905 df-f 4906 |
This theorem is referenced by: (None) |
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